MSRI Evans Talk: Dynamics on moduli spaces of flat surfaces - questions and new directions
Maryam Mirzakhani (Stanford University)
60 Evans Hall
In this talk, we will discuss various basic problems regarding the topology, geometry and dynamical systems on surfaces: all these questions are closely related to the study of closed Riemann surfaces endowed with a holomorphic one-form. This is the same as having a flat metric on the surface with finitely many cone-type singularities.
The moduli space of holomorphic one-forms on a closed surface of genus $g$ has a natural piecewise linear structure and carries an action of GL(2,R).
One can investigate the geometry and dynamics of an individual flat surface by studying its orbit under this linear group action.
One famous example is how understanding a billiard path on a rational billiard is related to studying the orbits of the action of GL(2,R) on the moduli spaces of flat surfaces.
Many of the results in this area are motivated by statements in homogenous dynamics. On the other hand, they have strong connections with other topics in algebraic geometry and number theory. In this talk, I will give an overview of some known results and open questions regarding these moduli spaces.